FLINT3.7.0-dev
/

acb_hypgeom_bessel_j_asymp

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_bessel_j_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec)source

Sets res to the Bessel function of the first kind Jν(z)J_\nu(z) computed with the asymptotic expansion of acb_hypgeom_u_asymp.

For all complex ν,z\nu, z,

Jν(z)=zν2νeizΓ(ν+1)1F1(ν+12,2ν+1,2iz)=A+B++A−B−J_{\nu}(z) = \frac{z^{\nu}}{2^{\nu} e^{iz} \Gamma(\nu+1)} {}_1F_1(\nu+\tfrac{1}{2}, 2\nu+1, 2iz) = A_{+} B_{+} + A_{-} B_{-}

where

A±=zν(z2)−12−ν(∓iz)12+ν(2π)−1/2=(±iz)−1/2−νzν(2π)−1/2,B±=e±izU∗(ν+12,2ν+1,∓2iz).A_{\pm} = z^{\nu} (z^2)^{-\tfrac{1}{2}-\nu} (\mp i z)^{\tfrac{1}{2}+\nu} (2 \pi)^{-1/2} = (\pm iz)^{-1/2-\nu} z^{\nu} (2 \pi)^{-1/2}, \qquad B_{\pm} = e^{\pm i z} U^{*}(\nu+\tfrac{1}{2}, 2\nu+1, \mp 2iz).

Simpler forms of A±A_{\pm} hold in special cases and are used when they apply: if ν+12=n∈Z\nu + \tfrac{1}{2} = n \in \mathbb{Z}, then A±=(±i)n(2πz)−1/2A_{\pm} = (\pm i)^{n} (2 \pi z)^{-1/2}, and if Re⁡(z)>0\operatorname{Re}(z) > 0, then A±=exp⁡(∓i[(2ν+1)/4]π)(2πz)−1/2A_{\pm} = \exp(\mp i [(2\nu+1)/4] \pi) (2 \pi z)^{-1/2}.

Cite this page

The FLINT team, “acb_hypgeom_bessel_j_asymp (acb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_hypgeom/acb_hypgeom_bessel_j_asymp.html (generated 2026-11-02).